Li Pi Su

Abstract

Let X and Y be any subsets of
En, and (X′,d1) and (Y ′,d2) be any metric spaces. Let Cm(X),0 ≦ m ≦∞,
denote the ring of m-differentiable functions on X, and Lc(X′) be the ring of
the functions which are Lipschitzian on each compact subset of X′, and
L(X′) be the ring of the bounded Lipschitzian functions on X′. The relations
between algebraic properties of Cm(X), (resp. Lc(X′) or L(X′) and the
topological properties of X (resp. X′) are studied. It is proved that if X and Y ,
(resp. (X′,d1) and (Y ′,d2)) are m-realcompact, (resp. Lc-real-compact or compact)
then Cm(X)≅Cm(Y ) (resp. Lc(X′)≅Lc(Y ′) or L(X′)≅L(Y ′) if and only
if X and Y are Cm-diffeomorphic (resp. (X′,d1) and (Y ′,d2) are Lc or
L-homeomorphic).